Finite-Approximate Controllability for Fractional Composite Relaxation Equations with Different Nonlocal Conditions

In this paper, the finite-approximate controllability for a class of fractional composite relaxation equations with different nonlocal conditions is discussed. Firstly, under the condition that the nonlocal term is compact, the existence of mild solutions to the equations is obtained by employing resolvent theory, the variational method, and Schauder’s fixed-point theorem. Moreover, under the assumption that the corresponding linear equation is approximately controllable, the fractional composite relaxation equation with the nonlocal condition is derived to be finite-approximately controllable. Furthermore, the existence of mild solutions and the finite-approximate controllability to the equations are considered for the weaker nonlocal problem. Finally, the example of nonlocal problem is provided to verify the feasibility of the results in this paper.